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Graph Modification of Bounded Size to Minor-Closed Classes as Fast as Vertex Deletion

Authors Laure Morelle , Ignasi Sau , Dimitrios M. Thilikos

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ACM Subject Classification
  • Theory of computation → Parameterized complexity and exact algorithms
Keywords
  • Graph modification problems
  • Parameterized complexity
  • Graph minors
  • Flat Wall theorem
  • Irrelevant vertex technique
  • Algorithmic meta-theorem
  • Parametric dependence
  • Dynamic programming

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Abstract

A replacement action is a function ℒ that maps each graph H to a collection of graphs of size at most |V(H)|. Given a graph class ℋ, we consider a general family of graph modification problems, called ℒ-Replacement to ℋ, where the input is a graph G and the question is whether it is possible to replace some induced subgraph H₁ of G on at most k vertices by a graph H₂ in ℒ(H₁) so that the resulting graph belongs to ℋ. ℒ-Replacement to ℋ can simulate many graph modification problems including vertex deletion, edge deletion/addition/edition/contraction, vertex identification, subgraph complementation, independent set deletion, (induced) matching deletion/contraction, etc. We present two algorithms. The first one solves ℒ-Replacement to ℋ in time 2^poly(k) ⋅ |V(G)|² for every minor-closed graph class ℋ, where poly is a polynomial whose degree depends on ℋ, under a mild technical condition on ℒ. This generalizes the results of Morelle, Sau, Stamoulis, and Thilikos [ICALP 2020, ICALP 2023] for the particular case of Vertex Deletion to ℋ within the same running time. Our second algorithm is an improvement of the first one when ℋ is the class of graphs embeddable in a surface of Euler genus at most g and runs in time 2^𝒪(k⁹) ⋅ |V(G)|², where the 𝒪(⋅) notation depends on g. To the best of our knowledge, these are the first parameterized algorithms with a reasonable parametric dependence for such a general family of graph modification problems to minor-closed classes.

Cite As Get BibTex

Laure Morelle, Ignasi Sau, and Dimitrios M. Thilikos. Graph Modification of Bounded Size to Minor-Closed Classes as Fast as Vertex Deletion. In 33rd Annual European Symposium on Algorithms (ESA 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 351, pp. 7:1-7:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025) https://doi.org/10.4230/LIPIcs.ESA.2025.7

Author Details

Laure Morelle
  • LIRMM, Université de Montpellier, CNRS, Montpellier, France
Ignasi Sau
  • LIRMM, Université de Montpellier, CNRS, Montpellier, France
Dimitrios M. Thilikos
  • LIRMM, Université de Montpellier, CNRS, Montpellier, France

Funding

  • Morelle, Laure: French project ELiT (ANR-20-CE48-0008-01)
  • Sau, Ignasi: French project ELiT (ANR-20-CE48-0008-01)
  • Thilikos, Dimitrios M.: ANR project GODASse ANR-24-CE48-4377, French-German Collaboration ANR/DFG Project UTMA (ANR-20-CE92-0027), and Franco-Norwegian project PHC AURORA 2024-25 (Projet n° 51260WL)

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